The Derivation of the Curvature Formulae
If we have an arc of a circle, the length of the arc is
the radius of the circle is
and the angle subtended by the arc is
then
and the radius
The curvature of the circle is
We can generalise this idea to find the curvature of a curve at any point on the curve by taking the limit as
and
If a curve is the graph of a twice differentiable function
then the curvature can be calculated from the formula
Proof:
so
Differentiate with respect to
using the chain rule:
(1)
If
is the length of a small piece of curve then
Substitute this into (1) to obtain
after some rearrangement. Take the magnitude of both sides to obtain
For a curve given in parametric coordinates the curvature is given by
